In this article current directions in solving Lovász’s problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given. © 2009 Elsevier B.V. All rights reserved. 1. Historical motivation In 1969, Lovász [59] asked whether every finite connected vertex-transitive graph has a Hamilton path, that is, a simple path going through all vertices, thus tying toge...

It is shown that the graph QQ) C obtained from the 7 -cube QQ) by deletion of a perfect Hamming code C has a spanning self-complementary subgraph which is edge-transitive but not vertextransitive and also extremal among all the cube subgraphs which are square-blocking and codeavoiding. Our work uses combinatorial techniques involving orientations on the Fano plane and the resulting Steiner trip...

In this paper, we propose a new type of digital signatures which is specifically designed for graph-based big data system. The properties of the proposed signatures are twofold. On one side it possesses the features of transitive signatures: One can sign a graph in such a way that, given two signatures on adjacent edges ði; jÞ and ðj; kÞ, anyone with public information can compute a signature o...

A transitive signature scheme allows a signer to publish a graph in an authenticated and cost-saving manner. The resulting authenticated graph is indeed the transitive closure of the graph constructed by edges which are explicitly signed by the signer. A property of the transitive signature scheme enables such scenario is called composability. Composability means that by knowing signatures on t...

In a previous work, the authors found new families of linear binary completely regular codes with the covering radius ρ = 3 and ρ = 4. In this paper, the automorphism groups of such codes are computed and it is proven that the codes are not only completely regular, but also completely transitive. From these completely transitive codes, in the usual way, i.e., as coset graphs, new presentations ...

The status of the project of classifying finite primitive distance-transitive graphs is surveyed. Particular attention is given to the structure of the proof of the recently obtained classification of finite primitive affine distance-transitive graphs. © 2005 Elsevier Ltd. All rights reserved.

It is essential to provide authentication on mobile nodes in group communication to ensure security and privacy. The nodes that are interested in participating in the group communication form Graphs (V, E). In this paper we authenticate the mobile nodes through transitive closure property of the graph in the routing phase of the On Demand Multicast Routing Protocol (ODMRP) that forms Transitive...

In this paper, we give two constructions of weakly distance-regular digraphs of girth 2, and prove that certain quotient digraph of a commutative weakly distancetransitive digraph of girth 2 is a distance-transitive graph. As an application of the result, we not only give some constructions of weakly distance-regular digraphs which are not weakly distance-transitive, but determine a special cla...

A circulant is a Cayley graph of a cyclic group Arc transitive circulants of square free order are classi ed It is shown that an arc transitive circulant of square free order n is one of the following the lexicographic product Kb or the deleted lexicographic Kb b where n bm and is an arc transitive circulant or is a normal circulant that is Aut has a normal regular cyclic subgroup